An oxygen tank is constructed of a right cylinder of height...

An oxygen tank is constructed of a right cylinder of height $y$ and radius $x$ with two hemispheres of radius $x$ mounted on the top and bottom of the cylinder. Express the volume of the cylinder as a function of two variables, $x$ and $y,$ find $V(10,2),$ and explain what this means.

Key Concepts

Composite Solids

This concept involves creating complex shapes by combining simpler geometric forms. In this problem, an oxygen tank is seen as a composite solid formed by attaching two hemispheres to a cylinder. Understanding composite solids is crucial because their overall volume is determined by summing the volumes of the individual components.

Volume of a Cylinder

The volume of a cylinder is calculated with the formula V = ?x²y, where x represents the radius of the circular base and y is the height of the cylinder. This formula is essential for determining the space inside cylindrical objects and is applied directly in the problem.

Volume of a Sphere (and Hemispheres)

A sphere’s volume is given by V = (4/3)?x³, and a hemisphere is exactly half of a sphere. Since the oxygen tank features two hemispheres, their combined volume equals the volume of one full sphere. This concept is key to accurately calculating the total volume of the tank beyond just the cylindrical portion.

Function of Multiple Variables

Expressing the total volume as a function V(x, y) highlights the dependency of volume on more than one dimension. Here, x and y are the independent variables that influence the overall volume. Such functions help in modeling real-world scenarios where a change in any one measurement can affect the outcome.

Evaluation and Interpretation

When the function V(x, y) is evaluated at specific values (for example, x = 10 and y = 2), it provides a numerical measurement of the volume of the oxygen tank. This step demonstrates the process of substituting given dimensions into the formula to obtain a concrete result, which in this context indicates the actual capacity of the tank.

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